Syntax Definition wcel 2167

Description: Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see mmset.html#class.

The purpose of introducing wff A e. B here is to allow us to prove the wel 2168 of predicate calculus in terms of the wceq 1364 of set theory, so that we do not overload the e. connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2183 for more information on the set theory usage of wcel 2167.

Hypotheses

Ref	Expression
wcel.cA	class A
wcel.cB	class B

Assertion

Ref	Expression
wcel	wff A e. B

This syntax is primitive. The first axiom using it is ax-13 2169.